From cb7c756e05af0382fc0a4e9faead081f20410b88 Mon Sep 17 00:00:00 2001 From: Florian Atteneder <florian.atteneder@uni-jena.de> Date: Fri, 23 Aug 2024 08:37:23 +0000 Subject: [PATCH] dg1d: Add Bsplines1 (https://git.tpi.uni-jena.de/dg/dg1d.jl/-/merge_requests/215) Continuation of !210 This adds a dispatch type `Bspline1` similar to `Bspline2`. Based on this we implement the necessary method to be able to use it in `DGElement`. I also fixed a subtle bug in the quadrature rules I used in some of the `Bspline` methods. That is, I accidentally used a LG rule with `Npts=2`, which corresponds to a `N=Npts-1=1` order rule that is exact for polynomials up to `2*N=2` order. But for `Bspline2` we encounter fourth order polynomials, so we should be using `Npts=3`. Also adds a new option to the `Mesh` parameters section: * `kind = "modal_bspline1"`: Utilize a 1st order modal Bspline Ansatz for the DG evolution. --- plot/{bspline2.jl => bspline.jl} | 92 +++++++-- src/ScalarEq/setup.jl | 2 +- src/bspline.jl | 192 +++++++++++++++++- src/dgelement.jl | 61 ++++-- src/parameters.jl | 11 +- .../modal_bspline2_advection_sine/output1d.h5 | Bin 114536 -> 114536 bytes .../substep_output.h5 | Bin 75944 -> 75944 bytes test/UnitTests/src/test_dgelement.jl | 38 +++- 8 files changed, 353 insertions(+), 43 deletions(-) rename plot/{bspline2.jl => bspline.jl} (56%) diff --git a/plot/bspline2.jl b/plot/bspline.jl similarity index 56% rename from plot/bspline2.jl rename to plot/bspline.jl index 75baad63..1ca886e6 100644 --- a/plot/bspline2.jl +++ b/plot/bspline.jl @@ -4,16 +4,20 @@ using GLMakie using LinearAlgebra +let +if true + # bs = dg1d.Bspline.Bspline2(5) x, _ = dg1d.LGL.rule(5) -bs = dg1d.Bspline.Bspline2(x) +bs = dg1d.Bspline.Bspline1(x) +Npts = bs.Npts xx = collect(range(-2.0,2.0,length=300)) -Bs = [ dg1d.Bspline.polynomial(xx[i], j, bs) for i in 1:length(xx), j in 1:bs.Npts+1 ] -∂Bs = [ dg1d.Bspline.derivative_polynomial(xx[i], j, bs) for i in 1:length(xx), j in 1:bs.Npts+1 ] +Bs = [ dg1d.Bspline.polynomial(xx[i], j, bs) for i in 1:length(xx), j in 1:Npts ] +∂Bs = [ dg1d.Bspline.derivative_polynomial(xx[i], j, bs) for i in 1:length(xx), j in 1:Npts ] -Vs = [ dg1d.Bspline.polynomial(bs.xs[2+i], j, bs) for i in 1:bs.Npts, j in 1:bs.Npts+1 ] +Vs = [ dg1d.Bspline.polynomial(bs.xs[2+i], j, bs) for i in 1:bs.Npts, j in 1:Npts ] # display(Vs) -∂Vs = [ dg1d.Bspline.derivative_polynomial(bs.xs[2+i], j, bs) for i in 1:bs.Npts, j in 1:bs.Npts+1 ] +∂Vs = [ dg1d.Bspline.derivative_polynomial(bs.xs[2+i], j, bs) for i in 1:bs.Npts, j in 1:Npts ] # display(∂Vs) M = dg1d.Bspline.mass_matrix(bs) @@ -21,7 +25,7 @@ S = dg1d.Bspline.stiffness_matrix(bs) ST = transpose(S) BB = similar(M); fill!(BB, 0.0) o = 2 -for i in 1:bs.Npts+1, j in 1:bs.Npts+1 +for i in 1:Npts, j in 1:Npts for k in 1:bs.Npts-1 xl, xr = bs.xs[o+k], bs.xs[o+k+1] Nil, Nir = dg1d.Bspline.polynomial(xl, i, bs), dg1d.Bspline.polynomial(xr, i, bs) @@ -73,42 +77,77 @@ lines!(ax, xx, PoU, label=L"\Sigma_n B_n", linestyle=:dot) axislegend(ax) display(fig) +end # if +end + +let +if false # # fn = x -> x < 0.15 ? -x+1 : 0.0 # # fn = x -> x < 0.0 ? -x : 0.0 -Npts = 4 +Npts = 5 xs, _ = dg1d.LGL.rule(Npts) xx = collect(range(extrema(xs)...,length=100)) fn = x -> x < -0.5 ? -x+1 : 0.0 fx = fn.(xs) +T = zeros(Float64, Npts+1, Npts) +T[1,1] = T[end,end] = 1.0 +for i in 2:Npts + T[i,i-1:i] .= 0.5 +end +pinvT = pinv(T) +# display(T) +# display(pinvT) + knots = vcat(xs[1], (xs[1:end-1].+xs[2:end])./2, xs[end]) -avg_fx = vcat(fx[1], (fx[1:end-1].+fx[2:end])./2, fx[end]) +# avg_fx = vcat(fx[1], (fx[1:end-1].+fx[2:end])./2, fx[end]) +avg_fx = T * fx bs = dg1d.Bspline.Bspline2(xs) Vs = dg1d.Bspline.vandermonde_matrix(knots, bs) -intrp_Bs = [ dg1d.Bspline.polynomial(xx[i], j, bs) for i in 1:length(xx), j in 1:bs.Npts+1 ] +intrpmat_Bs = dg1d.Bspline.vandermonde_matrix(xx, bs) A_lsqr = transpose(Vs)*Vs b_lsqr = transpose(Vs)*avg_fx +intrpmat_Bs_to_collocs = dg1d.Bspline.vandermonde_matrix(xs, bs) +pinv_intrpmat_Bs_to_collocs = pinv(intrpmat_Bs_to_collocs) +# display(intrpmat_Bs_to_collocs) +# display(pinv_intrpmat_Bs_to_collocs) +# display(pinv_intrpmat_Bs_to_collocs*intrpmat_Bs_to_collocs) +# display(intrpmat_Bs_to_collocs*pinv_intrpmat_Bs_to_collocs) # nodal Bspline nodal_Bs = Vs \ avg_fx -fn_nodal_Bs = intrp_Bs * nodal_Bs +fn_nodal_Bs = intrpmat_Bs * nodal_Bs # quasi-nodal Bspline quasi_nodal_Bs = avg_fx -fn_quasi_nodal_Bs = intrp_Bs * quasi_nodal_Bs +fn_quasi_nodal_Bs = intrpmat_Bs * quasi_nodal_Bs +# quasi-nodal Bspline data +quasi_nodal_Bs_samples = Vs * quasi_nodal_Bs +# corrected interpolateD +c_quasi_nodal_Bs_samples = copy(quasi_nodal_Bs_samples) +# c_quasi_nodal_Bs_samples[3] = 0 +c_quasi_nodal_Bs_samples[3] /= 10 +c_quasi_nodal_Bs = Vs \ c_quasi_nodal_Bs_samples +c_quasi_nodal_Bs = max.(c_quasi_nodal_Bs, 0.0) +c_quasi_nodal_Bs_samples = Vs * c_quasi_nodal_Bs +fn_c_quasi_nodal_Bs = intrpmat_Bs * c_quasi_nodal_Bs + +cc_quasi_nodal_Bs = copy(quasi_nodal_Bs) +cc_quasi_nodal_Bs[3] = c_quasi_nodal_Bs[3] +fn_cc_quasi_nodal_Bs = intrpmat_Bs * cc_quasi_nodal_Bs # corrected nodal Bspline c_nodal_Bs = max.(nodal_Bs, 0.0) -fn_c_nodal_Bs = intrp_Bs * c_nodal_Bs +fn_c_nodal_Bs = intrpmat_Bs * c_nodal_Bs # least-square Bspline lsqr_Bs = A_lsqr \ b_lsqr -fn_lsqr_Bs = intrp_Bs * lsqr_Bs +fn_lsqr_Bs = intrpmat_Bs * lsqr_Bs # corrected least-sqaure Bspline c_lsqr_Bs = max.(lsqr_Bs, 0.0) -fn_c_lsqr_Bs = intrp_Bs * c_lsqr_Bs +fn_c_lsqr_Bs = intrpmat_Bs * c_lsqr_Bs # nodal Lagrange intrp_La = dg1d.lagrange_interp_matrix(xx, xs) @@ -132,14 +171,27 @@ scatterlines!(ax, xs, fx, color=:orange, markersize=22, linestyle=:dash) lines!(ax, xx, fn_quasi_nodal_Bs, color=:green, label="quasi-nodal Bspline") scatterlines!(ax, knots, quasi_nodal_Bs, color=:green, markersize=14, linestyle=:dash) -# lines!(ax, xx, fn_c_nodal_Bs, color=:red, label="corrected nodal Bspline") -# scatterlines!(ax, knots, c_nodal_Bs, color=:red, markersize=10, linestyle=:dash) +scatter!(ax, knots, quasi_nodal_Bs_samples, color=:brown, markersize=10, + label="quasi-nodal Bspline samples") +scatter!(ax, knots, c_quasi_nodal_Bs_samples, color=:cyan, markersize=8, + label="corrected quasi-nodal Bspline samples") +lines!(ax, xx, fn_c_quasi_nodal_Bs, color=:magenta, label="corrected quasi-nodal Bspline") +scatterlines!(ax, knots, c_quasi_nodal_Bs, color=:magenta, markersize=6, linestyle=:dash) -lines!(ax, xx, fn_lsqr_Bs, color=:violet, label="least-square Bspline") -scatterlines!(ax, knots, lsqr_Bs, color=:violet, markersize=14, linestyle=:dash) +lines!(ax, xx, fn_cc_quasi_nodal_Bs, color=:limegreen, label="a posteriori corrected quasi-nodal Bspline") +scatterlines!(ax, knots, cc_quasi_nodal_Bs, color=:limegreen, markersize=6, linestyle=:dash) -lines!(ax, xx, fn_c_lsqr_Bs, color=:purple, label="corrected least-square Bspline") -scatterlines!(ax, knots, c_lsqr_Bs, color=:purple, markersize=14, linestyle=:dash) +lines!(ax, xx, fn_c_nodal_Bs, color=:red, label="corrected nodal Bspline") +scatterlines!(ax, knots, c_nodal_Bs, color=:red, markersize=6, linestyle=:dash) + +# lines!(ax, xx, fn_lsqr_Bs, color=:violet, label="least-square Bspline") +# scatterlines!(ax, knots, lsqr_Bs, color=:violet, markersize=14, linestyle=:dash) + +# lines!(ax, xx, fn_c_lsqr_Bs, color=:purple, label="corrected least-square Bspline") +# scatterlines!(ax, knots, c_lsqr_Bs, color=:purple, markersize=14, linestyle=:dash) axislegend(ax) display(fig) + +end # if +end diff --git a/src/ScalarEq/setup.jl b/src/ScalarEq/setup.jl index 7e85208e..3f906e69 100644 --- a/src/ScalarEq/setup.jl +++ b/src/ScalarEq/setup.jl @@ -15,7 +15,7 @@ function Project(env::Environment, mesh::Mesh1d, prms) analyze_error_norm = prms["ScalarEq"]["analyze_error_norm"] V_bernstein = dg1d.Bernstein.vandermonde_matrix(mesh.element.z) if mesh.element isa DGElement && mesh.element.kind === :modal_bspline2 - V_bspline = dg1d.Bspline.vandermonde_matrix(mesh.element.z, mesh.element.bs) + V_bspline = dg1d.Bspline.vandermonde_matrix(mesh.element.z, mesh.element.bs2) invV_bspline = pinv(V_bspline) else V_bspline, invV_bspline = nothing, nothing diff --git a/src/bspline.jl b/src/bspline.jl index 281336dc..96e5dd1c 100644 --- a/src/bspline.jl +++ b/src/bspline.jl @@ -5,6 +5,25 @@ using dg1d using Jacobi +struct Bspline1 + Npts::Int64 # number of unique knots + xs::Vector{Float64} # the knots, start and end point have multiplicity three +end + + +function Bspline1(z::Vector{Float64}) + Npts = length(z) + if Npts < 2 + error("z must contain at least two points, but you provided $Npts") + end + if any(d -> d < 0.0, diff(z)) + error("z must be a strictly increasing point set, but you provided $z") + end + xs = vcat(z[1], z, z[end]) + return Bspline1(Npts, xs) +end + + struct Bspline2 Npts::Int64 # number of unique knots xs::Vector{Float64} # the knots, start and end point have multiplicity three @@ -24,6 +43,31 @@ function Bspline2(z::Vector{Float64}) end +function polynomial(x::Float64, i::Int64, bs::Bspline1) + @unpack Npts, xs = bs + 1 ≤ i ≤ Npts || return 0.0 + @inbounds begin + xi, xi1, xi2 = xs[i], xs[i+1], xs[i+2] + end + if x < xi || x > xi2 + return 0.0 + end + if i == 1 && x == xi1 + # compute value on lhs bdry from the right + @goto sub2 + end + if x ≤ xi1 + return (x-xi)/(xi1-xi) + end + if x ≤ xi2 + @label sub2 + return (xi2-x)/(xi2-xi1) + end + # unreachable + return NaN +end + + function polynomial(x::Float64, i::Int64, bs::Bspline2) @unpack Npts, xs = bs 1 ≤ i ≤ Npts+1 || return 0.0 @@ -34,11 +78,11 @@ function polynomial(x::Float64, i::Int64, bs::Bspline2) return 0.0 end if i == 1 && x == xi2 - # compute derivative on lhs bdry from the right + # compute value on lhs bdry from the right @goto sub3 end if i == 2 && x == xi1 - # compute derivative on lhs bdry from the right + # compute value on lhs bdry from the right @goto sub2 end if x ≤ xi1 @@ -57,6 +101,31 @@ function polynomial(x::Float64, i::Int64, bs::Bspline2) end +function derivative_polynomial(x::Float64, i::Int64, bs::Bspline1) + @unpack Npts, xs = bs + 1 ≤ i ≤ Npts || return 0.0 + @inbounds begin + xi, xi1, xi2 = xs[i], xs[i+1], xs[i+2] + end + if x < xi || x > xi2 + return 0.0 + end + if i == 1 && x == xi1 + # compute derivative on lhs bdry from the right + @goto sub2 + end + if x ≤ xi1 + return 1/(xi1-xi) + end + if x ≤ xi2 + @label sub2 + return -1/(xi2-xi1) + end + # unreachable + return NaN +end + + function derivative_polynomial(x::Float64, i::Int64, bs::Bspline2) @unpack Npts, xs = bs 1 ≤ i ≤ Npts+1 || return 0.0 @@ -90,6 +159,40 @@ function derivative_polynomial(x::Float64, i::Int64, bs::Bspline2) end +""" + interpolate(x::Real, fn::AbstractVector, bs::Bspline1) + +Interpolate a Bspline polynomial with quasi-nodal coefficients `fn` onto `x`. +`x` is assumed to lie within `[-1,1]`. +The `fn` are assumed to be sampled on the nodes `xs = [ -1.0 + 2*i/N for i in 0:N ]` +where `N = length(fn)-1`. +""" +function interpolate(x::Real, coeffs::AbstractVector, bs::Bspline1) + @unpack Npts, xs = bs + if length(coeffs) != Npts + error("expected Npts = $Npts coefficients, received $(length(coeffs))") + end + return unsafe_interpolate(x, coeffs, bs) +end +function unsafe_interpolate(x::Real, coeffs::AbstractVector, bs::Bspline1) + @unpack Npts, xs = bs + @inbounds xl, xr = xs[1], xs[end] + xl == x && return coeffs[1] + xr == x && return coeffs[end] + xl < x < xr || return 0.0 + o = 1 # offset duplicated bdry knots + v_xs = view(xs, o .+ (1:Npts)) + ir = searchsortedfirst(v_xs, x) + il = ir-1 + is_supp = max(il-1,1):min(ir+1,Npts) + B = 0.0 + @inbounds for i in is_supp + B += coeffs[i] * polynomial(x, i, bs) + end + return B +end + + """ interpolate(x::Real, fn::AbstractVector, bs::Bspline2) @@ -124,6 +227,34 @@ function unsafe_interpolate(x::Real, coeffs::AbstractVector, bs::Bspline2) end +""" + mass_matrix(bs::Bspline1) + +Compute `M_ij = (Bi, Bj)` exactly where `Bi, Bj` are `i`th and `j`th Bspline of order 1. +""" +function mass_matrix(bs::Bspline1) + @unpack Npts, xs = bs + M = zeros(Float64, Npts, Npts) + zs, ws = dg1d.LG.rule(3) # 4th order accurate + o = 1 # offset bdry knots + for k in 1:Npts-1 + il, ir = o+k, o+k+1 + xl, xr = xs[il], xs[ir] + for i in 1:Npts, j in 1:Npts + mij = dg1d.LG.integrate(zs, ws) do z + x = ((xr-xl)*z + xr+xl)/2 + dxdz = (xr-xl)/2 + Bi = polynomial(x, i, bs) + Bj = polynomial(x, j, bs) + Bi*Bj*dxdz + end + M[i,j] += mij + end + end + return M +end + + """ mass_matrix(bs::Bspline2) @@ -132,7 +263,7 @@ Compute `M_ij = (Bi, Bj)` exactly where `Bi, Bj` are `i`th and `j`th Bspline of function mass_matrix(bs::Bspline2) @unpack Npts, xs = bs M = zeros(Float64, Npts+1, Npts+1) - zs, ws = dg1d.LG.rule(2) + zs, ws = dg1d.LG.rule(3) # 4th order accurate o = 2 # offset bdry knots for k in 1:Npts-1 il, ir = o+k, o+k+1 @@ -152,6 +283,34 @@ function mass_matrix(bs::Bspline2) end +""" + stiffness_matrix(bs::Bspline1) + +Compute `S_ij = (Bi, dBj/dx)` exactly where `Bi, Bj` are `i`th and `j`th Bspline of order 1. +""" +function stiffness_matrix(bs::Bspline1) + @unpack Npts, xs = bs + S = zeros(Float64, Npts, Npts) + zs, ws = dg1d.LG.rule(3) # 4th order accurate + o = 1 # offset bdry knots + for k in 1:Npts-1 + il, ir = o+k, o+k+1 + xl, xr = xs[il], xs[ir] + for i in 1:Npts, j in 1:Npts + sij = dg1d.LG.integrate(zs, ws) do z + x = ((xr-xl)*z + xr+xl)/2 + dxdz = (xr-xl)/2 + Bi = polynomial(x, i, bs) + ∂Bj = derivative_polynomial(x, j, bs) + Bi*∂Bj*dxdz + end + S[i,j] += sij + end + end + return S +end + + """ stiffness_matrix(bs::Bspline2) @@ -160,7 +319,7 @@ Compute `S_ij = (Bi, dBj/dx)` exactly where `Bi, Bj` are `i`th and `j`th Bspline function stiffness_matrix(bs::Bspline2) @unpack Npts, xs = bs S = zeros(Float64, Npts+1, Npts+1) - zs, ws = dg1d.LG.rule(3) + zs, ws = dg1d.LG.rule(3) # 4th order accurate o = 2 # offset bdry knots for k in 1:Npts-1 il, ir = o+k, o+k+1 @@ -180,11 +339,36 @@ function stiffness_matrix(bs::Bspline2) end +function vandermonde_matrix(z::AbstractVector, bs::Bspline1) + return [ Bspline.polynomial(z[i], j, bs) for i in 1:length(z), j in 1:bs.Npts ] +end function vandermonde_matrix(z::AbstractVector, bs::Bspline2) return [ Bspline.polynomial(z[i], j, bs) for i in 1:length(z), j in 1:bs.Npts+1 ] end +# analytical integration formula from +# Splines and PDEs: From Approximation Theory to Numerical Linear Algebra 2017, p. 13 +function integrate(f::AbstractVector, bs::Bspline1) + @toggled_assert length(f) == bs.Npts + sum = 0.0 + @inbounds for i in bs.Npts + fi, xl, xr = f[i], bs.xs[i], bs.xs[i+2] + sum += fi * (xr-xl)/2 + end + return sum +end +function integrate(f::Function, idxs::AbstractVector{<:Integer}, bs::Bspline1) + @toggled_assert length(idxs) == bs.Npts + sum = 0.0 + @inbounds for i in bs.Npts + fi, xl, xr = f[idxs[i]], bs.xs[i], bs.xs[i+2] + sum += fi * (xr-xl)/2 + end + return sum +end + + # analytical integration formula from # Splines and PDEs: From Approximation Theory to Numerical Linear Algebra 2017, p. 13 function integrate(f::AbstractVector, bs::Bspline2) diff --git a/src/dgelement.jl b/src/dgelement.jl index b5698cf4..3cd70700 100644 --- a/src/dgelement.jl +++ b/src/dgelement.jl @@ -29,7 +29,8 @@ struct DGElement V::Matrix{Float64} # = P_(j-1)(z_i), Vandermonde matrix invV::Matrix{Float64} # inv(V), inverse Vandermonde matrix F::Matrix{Float64} # spectral filtering matrix - bs::Bspline.Bspline2 # only set for :modal_bspline + bs1::Bspline.Bspline1 # only set for :modal_bspline1 + bs2::Bspline.Bspline2 # only set for :modal_bspline2 # TODO Replace N with Npts and remove N where ever possible. function DGElement(N::Integer, quadr::Symbol, kind::Symbol; @@ -39,13 +40,13 @@ struct DGElement if quadr ∉ (:lgl, :lgl_lumped, :glgl) error("Invalid quadrature rule specified for kind $kind: $quadr. Use one of lgl_lumped, lgl, glgl.") end - if kind ∉ (:nodal_lagrange, :modal_bernstein, :modal_bspline2) - error("Invalid DG formulation specified: $kind. Use one of nodal_lagrange, modal_bernstein, modal_bspline2.") + if kind ∉ (:nodal_lagrange, :modal_bernstein, :modal_bspline2, :modal_bspline1) + error("Invalid DG formulation specified: $kind. Use one of nodal_lagrange, modal_bernstein, modal_bspline2, modal_bspline1.") end Npts = N + 1 - local bs + local bs1, bs2 if kind === :nodal_lagrange if quadr === :lgl quadr_z, quadr_w = LGL.rule(Npts) @@ -91,6 +92,33 @@ struct DGElement l_rhs = zeros(Float64, Npts) l_lhs[1] = 1.0 l_rhs[end] = 1.0 + elseif kind === :modal_bspline1 + if N < 1 + error("Requested N+1=$(N+1) DOFs for DG formulation of kind $kind, but require at least 2.") + end + # User asked for Nth order polynomial sampled on Npts=N+1 points, + # we supply Npts 1st order Bsplines -- one for each DOF. + # bs_z is now the grid across which the Bsplines are C^0 continuous. + if quadr === :lgl || quadr === :lgl_lumped + z, _ = LGL.rule(Npts) + elseif quadr === :glgl + z, _ = GLGL.rule(Npts) + end + purge_zeros!(z) + bs1 = Bspline.Bspline1(z) + # Ignore given quadrature rule since we have to evaluate integrals on the subintervals, + # for which a 4th order accurate LG rule should suffice. + # We use a Gauss rule to avoid evaluating the Bspline on subcell interfaces and bdrys. + quadr_z, quadr_w = LG.rule(3) + D = [ Bspline.derivative_polynomial(z[i], j, bs1) for i in 1:Npts, j in 1:Npts ] + M = Bspline.mass_matrix(bs1) + S = Bspline.stiffness_matrix(bs1) + invM = inv(M) + V = Bspline.vandermonde_matrix(z, bs1) + l_lhs = zeros(Float64, Npts) + l_rhs = zeros(Float64, Npts) + l_lhs[1] = 1.0 + l_rhs[end] = 1.0 elseif kind === :modal_bspline2 if N < 2 error("Requested N+1=$(N+1) DOFs for DG formulation of kind $kind, but require at least 3.") @@ -105,7 +133,7 @@ struct DGElement bs_z, _ = GLGL.rule(N) end purge_zeros!(bs_z) - bs = Bspline.Bspline2(bs_z) + bs2 = Bspline.Bspline2(bs_z) # The dual grid of bs_z is the one on which we define the Npts quasi-nodal DOFs. # These are the nodes on which we evaluate residuals etc. z = vcat(bs_z[1], (bs_z[1:end-1].+bs_z[2:end])./2, bs_z[end]) @@ -113,21 +141,24 @@ struct DGElement # Ignore given quadrature rule since we have to evaluate integrals on the subintervals, # for which a 4th order accurate LG rule should suffice. # We use a Gauss rule to avoid evaluating the Bspline on subcell interfaces and bdrys. - quadr_z, quadr_w = LG.rule(2) - D = [ Bspline.derivative_polynomial(z[i], j, bs) for i in 1:Npts, j in 1:Npts ] - M = Bspline.mass_matrix(bs) - S = Bspline.stiffness_matrix(bs) + quadr_z, quadr_w = LG.rule(3) + D = [ Bspline.derivative_polynomial(z[i], j, bs2) for i in 1:Npts, j in 1:Npts ] + M = Bspline.mass_matrix(bs2) + S = Bspline.stiffness_matrix(bs2) invM = inv(M) - V = Bspline.vandermonde_matrix(z, bs) + V = Bspline.vandermonde_matrix(z, bs2) l_lhs = zeros(Float64, Npts) l_rhs = zeros(Float64, Npts) l_lhs[1] = 1.0 l_rhs[end] = 1.0 end + # just set up something, its unused for the other kinds + if kind !== :modal_bspline1 + bs1 = Bspline.Bspline1(z) + end if kind !== :modal_bspline2 - # just set up something, its unused for the other kinds - bs = Bspline.Bspline2(z) + bs2 = Bspline.Bspline2(z) end # TODO Rename MDM to MST or so, because the identity MDM = M^{-1} S^{T} only applies for nodal DG @@ -152,7 +183,7 @@ struct DGElement return new(N, Npts, kind, quadr, quadr_z, quadr_w, z, D, S, M, invM, MDM, Ml_lhs, Ml_rhs, - V, invV, F, bs) + V, invV, F, bs1, bs2) end end @@ -179,7 +210,7 @@ end @inline function integrate(u::AbstractVector, el::DGElement) - if el.kind === :modal_bspline2 + if el.kind ∈ (:modal_bspline2, :modal_bspline1) return Bspline.integrate(u, el.bs) else if el.quadr === :lgl || el.quadr === :lgl_lumped @@ -192,7 +223,7 @@ end @inline function integrate(fn::Function, idxs::AbstractVector{<:Integer}, el::DGElement) - if el.kind === :modal_bspline2 + if el.kind ∈ (:modal_bspline2, :modal_bspline1) return Bspline.integrate(fn, idxs, el.bs) else if el.quadr === :lgl || el.quadr === :lgl_lumped diff --git a/src/parameters.jl b/src/parameters.jl index 2160f97c..da44b181 100644 --- a/src/parameters.jl +++ b/src/parameters.jl @@ -516,6 +516,15 @@ end but the Bernstein polynomials are not interpolating. We refer to these coefficients as being quasi-nodal. See [2,3,4]. + - `modal_bspline1`: A modal approximation using 1st order Bspline polynomials. + The coefficients correspond to the nodal values sampled on the requested quadrature grid. + I.e. a `modal_bspline1` element has `n+1` DOF. + We adapt a quasi-nodal interpretation for the coefficients. + The bdry values correspond to the same bdry values as in a nodal Ansatz, + the cell interior values correspond to the cell-averages (determined by linear interpolation) + of a nodal Ansatz. + See [3,4]. + Note: Spectral filtering with the `filter_*` parameters is not implemented correctly. - `modal_bspline2`: A modal approximation using 2nd order Bspline polynomials. The coefficients correspond to the nodal values sampled on a grid that is dual (including the bdrys) to the grid of a `nodal_lagrange` Ansatz of order `n-1`. @@ -533,7 +542,7 @@ end [4] Kunoth et al., Splines and PDEs: From Approximation Theory to Numerical Linear Algebra (2018). """ dg_kind = "nodal_lagrange" - @check scheme in [ "nodal_lagrange", "modal_bernstein" ] + @check scheme in [ "nodal_lagrange", "modal_bernstein", "modal_bspline1", "modal_bspline2" ] """ Periodicity in cartesian directions `x,y` diff --git a/test/IntegrationTests/refs/modal_bspline2_advection_sine/output1d.h5 b/test/IntegrationTests/refs/modal_bspline2_advection_sine/output1d.h5 index d0e6758edad6c361ad5196a967dd777fb917170e..d960106c40355565a66ad8120a2c0e35f791f645 100644 GIT binary patch literal 114536 zcmeF)XIvD^+AjPC0Z{}oW5x_dFk*m0M8$-tm=hQ=f&p_F1LlC3FoGE~=1hSRMG;IO z2%=;}zyM;xb5-2~>RQ|P_wKXT-p^j=kq_KVPfc}o^?yQN(=#l$v+bZ?v_eru{OIZ` zT8g2fAurIcZ|WZxPD}GI=QnXu|B-*2W$Md*n)HVTdKA|gsh1m~fAJHiO$+R(D2_eb zw-<lr_n%+YSN|^m%V)r@eOm`H1^v|XpsTLIYG>+;1$JfQ{DuEQ{gq#f!qn++zyHGI z{x7=eulz1kKMv+kMgHUHU*%V`;E^|X39Vzr()@LaT2#?3xX;jXdVKz|jrdzb(V=$g z0cxoG)zF{?r`0#euZ3{ExTb_L!y3eowbE2iY5J{@yr5{S*Q=?iDLc?r_g6^6SdUux z{xmZs)xC(u$J+;wQj{6g?0@L@_j%2Krpm80pg+#5j{12|!%#zGhKWwWsPpIbkI#>8 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210ac199..c14d817b 100644 --- a/test/UnitTests/src/test_dgelement.jl +++ b/test/UnitTests/src/test_dgelement.jl @@ -1,4 +1,4 @@ -@testset "Spectral Element" begin +@testset "DGElement" begin function evaluate_derivatives(el::DGElement, f, df) @@ -202,7 +202,7 @@ end @test all(el.S.≈TST) el = dg1d.DGElement(N, :lgl, :modal_bspline2) - bs = el.bs + bs = el.bs2 M = el.M S = el.S ST = transpose(S) @@ -221,6 +221,26 @@ end dg1d.purge_zeros!(exact_BB) @test all(exact_BB .≈ BB) + el = dg1d.DGElement(N, :lgl, :modal_bspline1) + bs = el.bs1 + M = el.M + S = el.S + ST = transpose(S) + # verify the sum of surface terms + BB = dg1d.purge_zeros(S+ST) + exact_BB = similar(M); fill!(exact_BB, 0.0) + o = 1 + for i in 1:bs.Npts, j in 1:bs.Npts + for k in 1:bs.Npts-1 + xl, xr = bs.xs[o+k], bs.xs[o+k+1] + Nil, Nir = dg1d.Bspline.polynomial(xl, i, bs), dg1d.Bspline.polynomial(xr, i, bs) + Njl, Njr = dg1d.Bspline.polynomial(xl, j, bs), dg1d.Bspline.polynomial(xr, j, bs) + exact_BB[i,j] += Nir*Njr - Nil*Njl + end + end + dg1d.purge_zeros!(exact_BB) + @test all(exact_BB .≈ BB) + end @@ -267,6 +287,20 @@ end POU = dg1d.Bspline.interpolate.(xx, Ref(coeffs), Ref(bs)) @test all(POU .≈ ones(Float64, length(POU))) + + z, _ = dg1d.LGL.rule(5) + bs = dg1d.Bspline.Bspline1(z) + xx = collect(range(-1.0,1.0,length=25)) + Bs = [ dg1d.Bspline.polynomial(xx[i], j, bs) for i in 1:length(xx), j in 1:bs.Npts+1 ] + + ## partition of unity + POU = sum(Bs, dims=2) + @test all(POU .≈ ones(Float64, length(POU))) + # alternative test: evaluate a Bspline polynomial with all coefficients equal to one + coeffs = ones(Float64, bs.Npts) + POU = dg1d.Bspline.interpolate.(xx, Ref(coeffs), Ref(bs)) + @test all(POU .≈ ones(Float64, length(POU))) + end -- GitLab